2003 Apr

2003 April 23

Instructions. Each of the following \(7\) problems will be scored from \(0\) to \(10\). We us the following notations: \(\mathbb N = \{1, 2, 3, \dots\}\), \(\mathbb C\) is the set of complex numbers and \(\mathrm{Re}\ z\) denotes the real part of \(z\in \mathbb C\); \(D = \{z \in \mathbb C : |z| < 1\}\).

1. Does there exist a function having both of the following properties:\ (i) \(f\) is holomorphic on \(D\);\ (ii) \(\lim_{n\to \infty} |f(z_n)| = \infty\) co whenever \(\{z_n\}\) is a sequence of elements of \(D\) and \(\lim_{n\to \infty} |z_n| = 1\)?\ Justify your statement.

2. Does there exist sequence of functions \(\{f_n\}\) having all of the following properties:\ (i) \(f_n\) is holomorphic on \(D\) for all \(n \in \mathbb N\);\ (ii) \(|f_n(z)| < 1\) for all \(n \in \mathbb N\) and all \(z \in D\);\ (iii) \(\lim_{n\to \infty} |f_n(x)| = 0\) for all \(x \in (0,1)\);\ (iv) \(\lim_{n\to \infty} |f_n(x)| = 1\) for all \(x \in (-1,0)\).\ State a major theorem from complex analysis and use that theorem to justify your answer.

3. (a) Expand \(\sqrt{4-z^2}\) in a power series centered at \(0\). (The indicated function is to have the value \(2\) when \(z = 0\).)\ (b) Give the Laurent series expansion of \(\frac{1}{z(z-1)}\) in the region \(A = \{z \in \mathbb C : 2 < |z+2| < 3\}\).

4. Describe the image of the strip \(S = \{z \in \mathbb C : -\pi/2 < \mathrm{Re}\ z < \pi/2\}\) under the mapping \(z\mapsto \sin(z)\); include a description of the boundary correspondence.

5. Suppose that \(f\colon D \to \mathbb C\), \(f\) is holomorphic, \(|f(z)| \leq 1\) for all \(z \in D\), and \(f(1/2)=0\). Prove that \(|f(0)|\leq 1/2\).

6. State the homology version of the residue theorem. Use the residue theorem to prove that

   \[\int_0^\pi \frac{1}{4+\cos\theta}\ d\theta = \frac{\pi}{\sqrt{15}}.\]

7. Suppose that \((X, d)\) is a metric space, \(K\) is a compact subset of \(X\), and \(y \in X\). Prove that there exists a point \(p\in K\) such that \(d(p, y) \geq d(k, y)\) for all \(k \in K\).